Description: This figure replicates and extends the power curves reported in Kleibergen and Zhan (2020), Figures 3 and 4. It shows the Monte Carlo simulation-based rejection probability of the H0:λ= 2 for a consumption factor that prices the assets without a pricing error (left sub-figure, Z) or with pricing error (right sub-figure). Results are based on 10,000 draws of multivariate normally distributed data calibrated to the 31 test assets and unfiltered consumption (described in the caption of Table 4), with T = 55 and N = 31. Red/grey dotted is the GRS-test, reported as GRS-FAR test in Kleibergen and Zhan (2020). Red/grey is the actual GRS-FAR test. Black (dotted) shows the rejection frequency of the Fama-MacBeth t-statistics with the Shanken correction (when estimation is with an intercept). When moving along the x -axis, the test asset mean returns change according to the equation, $R¯˜t,ie=λ0+βC,iλ,$⁠, while the consumption betas remain the same.

Kleibergen and Zhan (2020) restrict their analysis to the case when the factor has no pricing errors (left sub-figure) and only show the area between the two vertical lines.

Interpretation: The Fama-MacBeth/Shanken approach with estimating the intercept leads to sharply increasing power curves just outside the area shown in Kleibergen and Zhan (2020). When the estimation is without the intercept, the power curve is comparable to the GRS-FAR test. Kleibergen and Zhan (2020) do not provide evidence of a “malfunction” of the Fama-MacBeth/Shanken method. Importantly, the GRS-FAR test cannot provide inference on the price of risk in the empirical relevant case when the risk factor comes with pricing errors (right sub-figure).